Triangle Penrose. Create an impossible triangle

Triangle Penrose. Create an impossible triangle
Triangle Penrose. Create an impossible triangle
13.06.2019

Head

mathematic teacher

1. Value ............................................................... 3

2. Historical reference ............................................. .. ... 4

3. Main part ..........................................................7

4. Proof of the impossibility of the triangle of Penrozov ... ... 9

5. Conclusions .................................................................... 11

6. The writer ....................................................... ...... 12

Relevance: Mathematics is an object that studies from the first graduation class. Many students consider it difficult, uninteresting and unnecessary. But if you look behind the page of the textbook, read additional literature, mathematical sophisms and paradoxes, then the idea of \u200b\u200bmathematics will change, the desire will have to study more than being studied in the school course of mathematics.

Purpose of work:

show that the existence of impossible figures will expand the horizons, develops spatial imagination, applied not only by mathematics, but also by artists.

Tasks :

1. To explore the literature on this topic.

2. Consider the impossible figures, make the model of the impossible triangle, to prove that the impossible triangle does not exist on the plane.

3. Make a scan of an impossible triangle.

4. Consider the examples of using the impossible triangle in the visual arts.

Introduction

Historically, mathematics played an important role in the visual arts, in particular when a perspective image, implying a realistic image of a three-dimensional scene on a flat canvas or a sheet of paper. According to modern views, mathematics and visual arts are very remote discipline, the first is analytical, the second is emotional. Mathematics does not play an obvious role in most works of contemporary art, and, in fact, many artists rarely or never even use the use of perspective at all. However, there are many artists who have mathematics in the spotlight. Several significant figures in the visual arts paved the road to these individuals.

In fact, there are no rules or restrictions on the use of various topics in mathematical art, such as, impossible figures, mebius tape, distortion or unusual perspective systems, as well as fractals.

History of impossible figures

Impossible figures are a certain kind of mathematical paradoxes consisting of regular parts connected in an irregular complex. If you try to formulate the definition of the term "impossible objects", it would probably sound like this - physically possible figures collected in the impossible form. But look at them is much more pleasant, drawing up definitions.

The errors of spatial construction met by artists and a thousand years ago. But the first to build and analyzing the impossible objects rightfully is considered to be Swedish artist Oscar RekersVard, who drawn in 1934. The first impossible triangle consisting of nine cubes.

Triangle ReutersVerda

Regardless of the RearSwerd, English Mathematician and physicist Roger Penrose re-opens the impossible triangle and publishes its image in the British magazine on psychology in 1958. In illusions used "false perspective". Sometimes such a perspective is called Chinese, since a similar pattern of drawing, when the depth of the "ambiguous" pattern, often met in the works of Chinese artists.

Escher waterfall

In 1961 Dutchman M. Escher, inspired by the impossible triangle of Penrose, creates a well-known lithography "Waterfall". Water in the picture flows infinitely, after the water wheel, it goes on and falls back to the starting point. In essence, this is an image of an eternal engine, but any attempt to construct this design is doomed to fail.

Another example of the impossible figures is presented in Figure "Moscow", which shows a not quite ordinary scheme of the Moscow Metro. First, we perceive the image entirely, but tracing individual lines with a look, we make sure of the impossibility of their existence.

« Moscow », Graphics (Mascara, Pencil), 50x70 cm, 2003

Figure "Three Snails" continues the tradition of the second famous impossible figure - the impossible cube (drawer).

"Three Snails" Impossible Cube

The combination of various objects can also be found in a not very serious figure "IQ" (intelligence coefficient). Interestingly, some people do not perceive the impossible objects due to the fact that their consciousness is not able to identify flat pictures with three-dimensional objects.

Donald Simanek expressed the opinion that the understanding of visual paradoxes is one of the signs of the type of creativity that the best mathematics, scientists and artists possess. Many work with paradoxical objects can be attributed to "Intellectual Mathematical Games". Modern science speaks of a 7-dimensional or 26-dimensional model of the world. You can simulate a similar world with the help of mathematical formulas, a person is simply not able to present it. And here are useful impossible figures.

The third popular impossible figure is an incredible staircase created by Penrose. You will continuously or climb (counterclockwise) or descend (clockwise). The Penrose model was based on the famous painting M. Escher "Up and Down" Incredible staircase Penrose

Impossible trident

"Damn fork"

There is another group of objects, which will not work. The classic figure is the impossible trident, or the "damn plug." With a careful study of the picture, it can be seen that three teeth gradually go into two on a single base, which leads to a conflict. We compare the number of teeth on top and bottom and come to the conclusion about the impossibility of the object. If you close the upper part of the trident, then we will see a completely real picture - three round teeth. If you close the lower part of the trident, we will also see the real picture - two rectangular teeth. But, if we consider the whole figure of the whole, it turns out that three round teeth are gradually turning into two rectangular.

Thus, it can be seen that the front and rear plans of this picture conflict. That is, what was originally in the foreground goes back, and the back plan (medium tooth) gets out forward. In addition to the change of the front and rear plans in this picture, there is another effect - the flat edge of the top of the trident becomes round at the bottom.

Main part.

Triangle- Figure consisting of 3 adjacent parts, which, with the help of unacceptable compounds of these parts, creates an illusion from a mathematical point of view of an impossible structure. Differently this three-point call is called Galnik Penrose

The graphic principle hiding behind this illusion is obliged to its formulation of a psychologist and his son Roger, physics. The salt of the Pen Frone consists of 3 squares of the square section located in 3 mutually perpendicular directions; Each is connected to the next right angle, all this is placed in three-dimensional space. Here is a simple recipe, how to draw this isometric projection of the salt of the Penfoors:

· Cut corners from the equilateral triangle on the lines parallel to the parties;

· Spend inside the cropped triangle parallels to the sides;

· Cut the corners again;

· Take another parallel inside;

· Imagine in one of the corners of some of the two possible cubes;

· Continue its L - shaped "thing";

· Run this design in a circle.

· If we chose another cube, the square would be "spinning" in the other way .

The scan of the impossible triangle.


Line of inflection

Cut line

What elements are the impossible triangle? More precisely, what elements he seems to us (it seems!) Built? The construction is based on a rectangular corner, which is obtained by a compound at a right angle of two identical rectangular bars. There are three pieces of such corners, and the bars, it became six pieces. These corners must be specifically "to connect" to one with another so that they formed a closed chain. What happens is the impossible triangle.

The first corner is placed in a horizontal plane. At this, connect the second corner, sending one of his ribs up. Finally, to this second corner to attach the third corner so that its edge is parallel to the original horizontal plane. At the same time, the two ribs of the first and third corners will be parallel and directed in different directions.

Now let's try to look at the figure from different points of space (or make a real wire layout). Imagine how it looks out of one point, from the other, from the third ... When a change in the observation point (or is that the same thing - when you turn the design in space) it will seem that two "end" ribs of our corners move relative to each other. It is not difficult to choose this position in which they connect (of course, at the same time the near corner will seem thicker than longer).

But if the distance between the ribs is much less than the distance from the corners to the point, from which we consider our design, then both ribs will have the same thickness for us, and there will be an idea that these two ribs are actually a continuation of each other.

By the way, if we at the same time look at the display of the design in the mirror, then I will not see a closed chain there.

And from the selected observation point, we have our own eyes with a challenge miracle: there is a closed chain of three corners. Just do not change the observation point so that this illusion (in fact it is the illusion!) Not collapsed. Now you can draw a visible object or put the camera lens in the found point and get a photo of the impossible object.

Penrose was interested in this phenomenon. They used the capabilities that occur when the three-dimensional space and three-dimensional objects are displayed on the two-dimensional plane (that is, when designing) and drew attention to some design uncertainty - an unlocked design of three-corners can be perceived as a closed chain.

As already mentioned, from the wire you can easily make the simplest model, in principle the explanatory observed effect. Take a straight piece of wire and divide it into three equal parts. Then bend the extreme parts so that they formed a straight angle with the middle part, and turn each other relative to another 900. Now turn this figure and watch it with one eye. With some of her position it will seem that it is formed from a closed piece of wire. When turning on the table lamp, you can watch the shadow falling on the table, which also in a certain arrangement of the shape in space turns into a triangle.

However, this feature of the design can be observed in another situation. If you make a ring from a wire, and then it is breeding it in different directions, then one turn is a cylindrical spiral. This turn, of course, open. But when designing it to the plane, you can get a closed line.

We once again were convinced that according to the projection on the plane, in the figure, the three-dimensional figure is restored ambiguously. That is, in the projection, some ambiguity, inexpensive, which generate the "impossible triangle" is enclosed.

And we can say that the "impossible triangle" of Penrose, as many other optical illusions, stands in one row with logical paradoxes and kalamboras.

Proof of the impossibility of the triangle of Penrose

Analyzing the features of a two-dimensional image of three-dimensional objects on the plane, we understood how the features of this mapping lead to an impossible triangle.

Prove that the impossible triangle does not exist, it is extremely easy, because every corner is direct, and their amount is equal to 2700 instead of "put" 1800.

Moreover, even if we consider the impossible triangle glued from the corners smaller than 900, then in this case you can prove that the impossible triangle does not exist.

Consider another triangle that consists of several parts. If the parts from which it consists, to arrange differently, it will turn out exactly the same triangle, but with one small flaw. Will not have enough single square. How is this possible? Or after all, this is an illusion.

https://pandia.ru/text/80/021/images/image016_2.jpg "alt \u003d" (! Lang: Impossible triangle" width="298" height="161">!}

Using phenomenon perception

Is it possible to somehow strengthen the effect of the impossibility? "Is it impossible" Are there any objects than others? And here the features of human perception come to the rescue. Psychologists have been established that the eye begins to inspect the object (pattern) from the left left corner, then the glance slides right to the center and falls into the lower right corner of the picture. Such a trajectory may be due to the fact that our ancestors at a meeting with the enemy first looked at the most dangerous right hand, and then the glance moved to the left, on the face and figure. Thus, artistic perception will significantly depend on how the composition of the painting is built. This feature in the Middle Ages is clearly manifested in the manufacture of tapestries: their drawing was a mirror reflection of the original, and the impression that tapestries and originals produce.

This property can be successfully used when creating creations with impossible objects, increasing or reducing the "degree of impossibility". The prospect of obtaining interesting compositions using computer technologies or from several paintings turned (maybe using a different type of symmetry) is one relative to another creating different impression from the viewers from the object and a deeper understanding of the essence of the plan, or from one, rotating ( Constantly or jerks) with a simple mechanism for some angles.

This direction can be called polygonal (polygonal). There are images that turned one relative to the other. The composition was created as follows: the drawing on paper, made in mascara and a pencil, was scanned, was translated into a digital form and processed in a graphic editor. It can be noted a pattern - the rotated picture has a greater "degree of impossibility" than the initial one. This is easily explained: the artist during the work subconsciously seeks to create a "correct" image.

Conclusion

The use of various mathematical figures and laws is not limited to the above examples. Carefully studying all the above figures, you can find other not mentioned in this article, geometric bodies or visual interpretation of mathematical laws.

Mathematical art flourishes today, and many artists create paintings in the style of Escher and in their own style. These artists work in various directions, including sculpture, drawing on flat and three-dimensional surfaces, lithography and computer graphics. And the most popular topics of mathematical art remain polyhedra, impossible figures, mebius tapes, distorted perspective systems and fractals.

Conclusions:

1. So, the consideration of impossible figures develop our spatial imagination, help "exit" from the plane into three-dimensional space, which will help when studying stereometry.

2. Models of impossible figures help consider projections on the plane.

3. Consideration of mathematical sophishes and paradoxes are instilled in mathematics.

When performing this work

1. I found out - how, when, where, and by whom I was first considering the impossible figures that there were many such figures, these figures are constantly trying to portray artists.

2. I, together with the dad, made a model of an impossible triangle, considered its projection on the plane, saw the paradox of this figure.

3. considered the reproductions of the artists in which these figures are depicted

4. My research was interested in classmates.

In the future, the knowledge gained I will use in mathematics lessons and interested me, but do other paradoxes exist?

LITERATURE

1. Candidate of Technical Sciences D. Rakov History of impossible figures

2. Ruetevard O. Impossible figures. - M.: Stroyzdat, 1990.

3. Website V. Alekseev Illusion · 7 Comments

4. J. Timothy Anrachi. - Amazing figures.
(LLC "Publisher AST", LLC "ASTELL PUBLISHER", 2002, 168 p.)

5. . - Graphics.
(Art Rodnik, 2001)

6. Douglas Hofstadter. - Goodles, Escher, Bach: This endless garland. (Publishing House "Bakhrakh-M", 2001)

7. A. Konenko - Secrets of impossible figures
(Omsk: Lefty, 199)


Impossible is still possible. And bright confirmation of this is the impossible triangle of Penrose. Open in the last century, it is often often found in the scientific literature. And no matter how surprisingly it sounds, but it can be made even on your own. And this is completely easy to do it. Many lovers draw or collect origami have long been able to do it.

The value of the triangle Penrose

There are several names of this figure. Some call it the impossible triangle, others - just tribar. But most often you can find the definition of the "Triangle of Penrose".

Understand these definitions one of the main impossible figures. If you judge the name, then it is impossible to get a similar figure in reality. But in practice it was proved that it is still possible to do. That's just the form will take if you look at it from a certain point below the desired angle. From all other sides, the figure is quite real. It represents three edges of Cuba. And make a similar design easily.

History opening

Penrose's triangle was opened back in 1934 by the artist from Sweden Oscar Reethersvard. The figure was presented in the form of cubes collected together. In the future, the artist began to call the "father of impossible figures."

Perhaps the drawing of Reetherswend so remains little-known. But in 1954, the Swedish mathematician Roger Penrose wrote an article about impossible figures. It became the second birth of a triangle. True, the scientist presented it in more familiar. He used not cubes, but beams. Three beams were connected to each other at an angle of 90 degrees. The difference was that Reuthersvard used the parallel perspective during drawing. And Penrose applied a linear perspective, which gave the picture even more inability. Such a triangle was published in 1958 in one of the British magazines on psychology.

In 1961, the artist Mauritz Escher (Holland) created one of its most popular lithographs "Waterfall". It was created under the impression that was caused by an article about impossible figures.

In the eighties of the last century, the tribar and other impossible figures were depicted on the state postage stamps of Sweden. It continued for several years.

At the end of the last century (or rather in 1999), an aluminum sculpture was created in Australia, depicting the impossible triangle of Penrose. It reached a height of 13 meters. Such sculptures, only smaller in size, are found in other countries.

Impossible in reality

As it was possible to guess, the triangle of Penrose is not really a triangle in the usual understanding. It is three faces of Cuba. But if you look from a certain angle, the illusion of the triangle is obtained due to the fact that 2 corners completely coincide on the plane. Spearly aligning the closest from the looking and far corner.

If you are attentive, you can guess that the tribar is nothing more than illusion. The actual look of the figure can give the shadow from her. It shows it that in fact the angles are not connected. Well, of course, everything becomes clear if the shape takes into hand.

Making a figure with their own hands

Penrose triangle can be collected independently. For example, paper or cardboard. And help in this scheme. They only need to print and glue. On the Internet there are two schemes. One of them is a little easier, the other is more complicated, but more popular. Both are presented in the drawings.

Penrose's triangle will become an interesting product that will definitely enjoy guests. He will definitely be unnoticed. The first step to create it is the preparation of the scheme. It is transferred to paper (cardboard) with a printer. And then it's still easier. It you just need to cut around the perimeter. The diagram already has all the necessary lines. It will be more convenient to work with more dense paper. If the scheme is printed on fine paper, but I want something more slowly, the billet is simply applied to the selected material and cut down the contour. So that the scheme does not shift, it can be attached with paper clips.

Next, you need to define those lines by which the workpiece will bend. As a rule, in the diagram it is represented by bending the item. Next, we determine the places that are subject to gluing. They are missing PVA glue. The item is connected to a single figure.

Detail can be painted. And you can initially use color cardboard.

Draw an impossible figure

Penrose triangle can also be drawn. To begin with, a simple square is drawn on the sheet. Its size does not matter. Based on the lower side of the square, a triangle is drawn. In his corners, small rectangles are drawn inside. Their parties will need to erase, leaving only those that are common with a triangle. As a result, a triangle with truncated angles should be turned out.

On the left side of the upper lower angle there is a straight line. The same line, but a little shorter, is drawn from the leftmost corner. In parallel, the base of the triangle is carried out a line coming out of the right corner. It turns out the second dimension.

According to the principle of the second, the third dimension is drawn. Only in this case, all direct are based on the angles of the figure not the first, but the second dimension.

A few impossible figures are invented - a staircase, a triangle and a X-teeth. These figures are actually quite real in the volume image. But when the artist designs the volume on paper, objects seem impossible. The triangle, which is still called the Tribar, has become a wonderful example of how impossible becomes possible when you apply efforts.

All these figures are beautiful illusions. Achievements of human genius use artists who draw in the style of impressed.

Nothing is impossible. So we can say about the triangle of Penrose. This is a geometrically impossible figure, the elements of which cannot be connected. Still, the impossible triangle became possible. The Swedish painter Oscar Reethersward in 1934 presented the world an impossible triangle from cubes. O. Reethersward is considered the discoverer of this visual illusion. In honor of this event, this drawing was published on the Postage Mark Sweden later.

And in 1958, a publication in the English magazine about impossible figures was published by Mathematics Roger Penrose. It was he who created a scientific model of illusion. Roger Penrose was an incredible scientist. He conducted research in the theory of relativity, as well as an exciting quantum theory. He was awarded Wolf Prize together with S. Hawking.

It is known that the artist Mauritz Escher, being impressed by this article, painted his amazing work - the lithograph of the "Waterfall". But is it possible to make a triangle of Penrose? How to do, if possible?

Trybar and reality

Although the figure is considered impossible, make a triangle of Penroise with his own hands - easier simple. It can be made of paper. Origami lovers simply could not get around the Tribar's side and found the way to create and hold the thing in their hands, which seemed earlier the scholars' fantasy.

However, our own eyes are deceived, when we look at the projection of the three-dimensional object of three perpendicular lines. The observer seems to see the triangle, although in fact it is not so.

Geometry crafts

Triangle Tribar, as stated, is actually a triangle is not. Triangle Penrose - Illusion. Only at a certain angle, the object looks like an equilateral triangle. However, the object in natural form is 3 faces of the cube. On such an isometric projection coincide on the plane 2 of the corner: the closest from the viewer and the far.

Optical deception, of course, is quickly revealed, only only take this object in hand. And also reveals the illusion of the shadow, since the shadow of the tribara clearly shows that the angles do not coincide in reality.

Tribar of paper. Schemes

How to make a triangle of Penroise with her hands from paper? Are there any diagrams of this model? Today, 2 scrolls are invented in order to fold such an impossible triangle. The basics of geometry suggest exactly how to put the object.

To fold the triangle of Penroise with her own hands, it will be necessary to allocate only 10-20 minutes. It is necessary to prepare glue, scissors for several cuts and paper on which the scheme is printed.

From such a workpiece, the most popular impossible triangle is obtained. Craft-origami is not too complicated in the manufacture. Therefore, it is imperative from the first time, and even at a schoolboy, just started studying geometry.

As you can see, it turns out a very pretty handicraft. The second blank looks different and folds differently, but the triangle of Penrose itself looks the same.

Stages of creating a pen-penal triangle from paper.

Choose one of 2 convenient blanks for you, copy the file and print. Here is an example and the second model of the scroll, which is performed slightly easier.

The billet for the origami "Tribar" already contains all the necessary prompts. In fact, the instruction for the scheme is not required. Just download to a dense paper medium, otherwise it will not work uncomfortable and the figure will not work. If you can not immediately print on the cardboard, you need to make a sketch to a new material and the contour cut the drawing. For convenience, you can cure paper clips.

What to do then? How to fold the triangle of Penroise with her hands in stages? You need to follow this plan of action:

  1. We carry out the opposite side of the scissors those line where you need to bend, according to the instructions. Bend all lines
  2. Where necessary, make cuts.
  3. We glue with the help of PVA those Loskutka, which are designed to fasten the parts into a single integer.

The finished model can be repainted in any color, or in advance to work with color cardboard. But even if the object is from white paper, anyway, everyone who enters your living room for the first time will certainly be discouraged by such a cradle.

Drawing triangle

How to draw a triangle Penrose? Not everyone loves to engage in origami, but many love to draw.

To begin with, the usual square of any size is depicted. Then the triangle is drawn inside, the basis of which is the bottom side of the square. A small rectangle fits into each corner, all sides of which are erased; Only those parties remain to be adjacent to the triangle. It is necessary that the lines are smooth. It turns out a triangle with truncated corners.

The next step is the image of the second dimension. From the left side of the upper lower corner, a strictly straight line is carried out. The same line is carried out, starting from the lower left corner, and is not a bit brought to the first line 2 of the measurement. Another line is drawn from the right corner parallel to the bottom of the main figure.

The final stage - inside the second dimension is drawn third with three small lines. Small lines start from the second dimension lines and the image of the three-dimensional volume is completed.

Other Figures Penrose

Upon the same analogy, other figures can be drawn - a square or a hexagon. Illusion will be observed. But still these figures are no longer so shook imagination. Such polygons seem just very twisted. Modern graphics makes it possible to make more interesting versions of the famous triangle.

In addition to the triangle, the Ladder of Penrose is also known world famous. The idea is to deception when it seems that a person rises continuously up when driving clockwise, and if moving counterclockwise, then down.

The continuous staircase is known more than the Association with the picture M. Escher "Climbing and Descent". Interestingly, when a person passes all 4 spans of this illusory staircase, he invariably turns out there, from where he started.

Other objects introducing a person misconception are known, such as the impossible bar. Or made according to the same laws of illusion box with intersecting faces. But all these objects are already invented on the basis of the article of a wonderful scientist - Roger Penrose.

Impossible triangle in pep

Figure, named after mathematics, was honored. She has a monument. In 1999, a large triangle of Penrose from aluminum was installed in one of the cities of Australia (Perth), which is 13 meters in height. Near the aluminum giant, tourists are glad to be photographed. But if you choose another angle view for the photo, the deception becomes obvious.

Greetings to you dear blog readers Website. In touch Rustam Zakirov and I have another article for you, the topic of which to draw a triangle of Penrose. Today I want to show you how it is easy and simply can draw an impossible triangle. We will draw two pictures of this triangle, one will be usual, and the second is the real 3D drawing. And all this will be surprisingly simple. The present 3D drawing of this triangle you can. I doubt that this will be shown somewhere else, so read the article to the end and very carefully.

For our drawings, we will need as always: a piece of paper. Simple pencils (preferably one "middle", "other soft") and several colored pencils or markers.

How easy it is to draw any 3D drawings.

I pulled out this impossible triangle here from this ordinary picture, which I just found on the Internet. Here she is.

And then in a couple of minutes with the help transferred it to 3D . This can be translated into 3D almost any images. Who wants to learn the same, click here.

And we go to our drawing.

Draw a regular triangle drawing.

Step number 1. Transfer from the screen of the monitor.

In order to draw a triangle to you, you will need to do the following. You take your sheet of paper and lean it to the triangle on the monitor screen, and just translate it.

And since our triangle is not completely difficult, just put only the main points in all its corners.

And then we look at the original and connect these points using the ruler. I got like this.

All our triangle is ready. You can leave like this, but let us show it a little later. I did it with colored pencils. After we completely decorated our triangle, once again we completely supply it with a simple soft pencil.

On this, our ordinary triangle Penrose is completely ready, and we move to the same triangle.

We draw a 3D drawing of a triangle.

Step number 1. Translate.

We act on the same scheme as with a conventional pattern. I give you a ready-made triangle format already translated into 3D. Here it is.

And you translate it. We do everything as well as with a regular pattern. You take your sheet, lean it back to the monitor screen, the leaflet shines, and you simply translate the ready 3D drawing to your sheet.

That's what happened to me.

Triangle size can be increased or decreased. To do this, you just need to change the scale of your monitor. Hold down the Ctrl key and turn the mouse wheel.

We can safely say that our 3D drawing is ready. I went to him about 3 minutes. On this, in principle, you can safely finish, but let's decoke on our triangle.

Dmitry Rakov

Our eyes do not know how
Native objects.
And therefore do not impose them
delusions of reason.

Tit Lucretia Car

The situation "illustrate" is incorrectly incorrect. Eyes cannot deceive us, because they are only an intermediate link between the object and the human brain. The imprisonment usually arises not because we see, but due to the fact that they unconsciously argue and unwittingly mistaken: "Through the eye, and not an eye to look at the world knows how to look."

One of the most spectacular directions of the artistic flow of optical art (OP-ART) is an IMP-ART (IMP-ART, IMPOSSIBLE ART), based on the image of the impossible figures. The impossible objects are drawings on the plane (any plane of two-dimensional), depicting three-dimensional structures, the existence of which in the real three-dimensional world is impossible. Classical and one of the simplest figures is the impossible triangle.

In the impossible triangle, every angle itself is possible, but the paradox occurs when we consider it entirely. The sides of the triangle are directed simultaneously to the viewer, and from it, therefore its individual parts cannot form a real three-dimensional object.

Actually, our brain interprets the pattern on the plane as a three-dimensional model. Consciousness sets the "depth" on which every point of the image is. Our ideas about the real world are faced with a contradiction with some inconsistency, and have to do some assumptions:

  • direct two-dimensional lines are interpreted as direct three-dimensional lines;
  • two-dimensional parallel lines are interpreted as three-dimensional parallel lines;
  • sharp and stupid angles are interpreted as direct angles in the future;
  • external lines are considered as a border of the form. This external border is extremely important for building a full image.

Human consciousness first creates a general image of the subject, and then examines individual parts. Each angle is compatible with a spatial perspective, but, reunited, they form a spatial paradox. If you close any of the corners of the triangle, then the inability disappears.

History of impossible figures

The errors of spatial construction met by artists and a thousand years ago. But the first to build and analyzing the impossible objects rightfully is considered to be Swedish artist Oscar ReuthersVard (Oscar Reutersvärd), who drew the first impossible triangle in 1934, consisting of nine cubes.

"Moscow", graphics
(mascara, pencil),
50x70 cm, 2003

Regardless of the RearSwerd, English Mathematician and physicist Roger Penrose re-opens the impossible triangle and publishes his image in the British magazine on psychology in 1958. In the illusion used "false perspective". Sometimes such a perspective is called Chinese, since a similar pattern of drawing, when the depth of the "ambiguous" pattern, often met in the works of Chinese artists.

The figure "Three Snails" small and large Cuba is not oriented in a normal isometric projection. The smaller cube sizes conjugates with large front and rearrangements, and therefore following three-dimensional logic, it has the same dimensions of some parties as large. First, the drawing seems to be a real representation of a solid body, but as the logical contradictions of this object are revealed as it is analyzed.

Figure "Three Snails" continues the tradition of the second famous impossible figure - the impossible cube (drawer).

"IQ", graphics
(mascara, pencil),
50x70 cm, 2001 		"Up and down",
M. Escher

The combination of various objects can be found in a not very serious figure "IQ" (Intelligence QUotient - the intelligence coefficient). Interestingly, some people do not perceive the impossible objects due to the fact that their consciousness is not able to identify flat pictures with three-dimensional objects.

Donald E. Simandek expressed the opinion that an understanding of visual paradoxes is one of the signs of the type of creative potential that the best mathematics, scientists and artists possess. Many work with paradoxical objects can be attributed to "Intellectual Mathematical Games". Modern science speaks of a 7-dimensional or 26-dimensional model of the world. You can simulate a similar world with the help of mathematical formulas, a person is simply not able to present it. And here are useful impossible figures. From a philosophical point of view, they serve as a reminder that any phenomena (in systemic analysis, science, politics, economics, etc.) should be considered in all complex and non-obvious relationships.

A variety of impossible (and possible) objects are presented in the picture "Impossible alphabet".

The third popular impossible figure is an incredible staircase created by Penrose. You will continuously or climb (counterclockwise) or descend (clockwise). The Penrose model was based on the famous painting M. Escher "Up and Down" ("Ascending and Descending").

There is another group of objects, which will not work. The classic figure is the impossible trident, or the "damn plug".

With a careful study of the picture, it can be seen that three teeth gradually go into two on a single base, which leads to a conflict. We compare the number of teeth on top and bottom and come to the conclusion about the impossibility of the object.

Is there any more significant benefit from impossible drawings than the mind of the mind? In some hospitals, the images of the impossible objects specifically hang up, since their viewing is capable of occupying patients for a long time. It would be logical to raise such drawings at the box office, in militia and other places, where the waiting for its turn lasts sometimes the whole eternity. Figures could act as such "chronophages", i.e. Eaters of time.